Skyrmion in spinor condensates and its stability in trap potentials

A necessary condition for the existence of a skyrmion in two-component Bose-Einstein condensates with SU(2) symmetry was recently provided by two of the authors [I. F. Herbut and M. Oshikawa, Phys. Rev. Lett. 97, 080403 (2006)], by mapping the problem to a classical particle in a potential subject to time-dependent dissipation. Here we further elaborate this approach. For two classes of models, we demonstrate the existence of the critical dissipation strength above which the skyrmion solution does not exist. Furthermore, we discuss the local stability of the skyrmion solution by considering the second-order variation. A sufficient condition for the local stability is given in terms of the ground-state energy of a one-dimensional quantum-mechanical Hamiltonian. This condition requires a minimum number of bosons, for a certain class of the trap potential. In the optimal case, the minimum number of bosons can be as small $\sim {10}^4$.

Figure 1

(Color online) The ratio $-\left(dy/dt\right)/y$ as a function of time $t$, within constant-$\eta$ equation (33) and with boundary conditions (26). In the actual numerical calculation, a small initial velocity is given instead of zero in Eq. (26) to avoid getting only the trivial solution. The result is robust against the change in the small initial velocity. In the figure, we show the result for $\eta =4$ as example. The ratio $-\left(dy/dt\right)/y$ increases monotonically, asymptotically approaching ${\lambda }_1=2-\sqrt{2}\sim 0.585786$. We obtained similar results for other $\eta >2\sqrt{2}$.

Figure 2

(Color online) Trajectories of $\overline{\omega }\left(t\right)$ in the case of $\eta \left(t\right)=-n\text{tanh}t$. The trajectories in $t<0$ are antisymmetric to that in $t>0$, so we show $\overline{\omega }\left(t\right)$ only for $t>0$. Here, calculating the trajectories numerically, $\overline{\omega }\left(t=0\right)=\pi /2$ is imposed as an initial condition. Now, although we define $n>2$, the trajectory for $n=2$ is also shown.

Figure 3

The velocity of the particle at $t=0$ as a function of $n_c-n$, for the tanh dissipation [Eq. (31)]. The squares are numerical results obtained by the shooting method. The line is the best fit assuming a power law. The critical parameter $n_c$ is determined by the analytic prediction. The good fit to the power law means that the analytic prediction of $n_c$ is consistent with the numerical calculation. Moreover, the exponent $0.979\cdots$ obtained by the fit is also consistent with the analytic prediction of unity .

Figure 4

The velocity of the particle at the potential minimum $\left(t=0\right)$ as a function of $n_c-n$, for the step-function dissipation. The line is the best fit assuming a power law to the numerical results, which are shown as squares. The critical parameter $n_c$ is determined by the analytic prediction. The numerical result is again in good agreement with the predictions.

Figure 5

(Color online) Plots of the effective potential $V_{\text{eff}}$ defined in Eq. (23) for given $\eta \left(t\right)=-n\text{tanh}t$. In the left graph (a), $V_{\text{eff}}\left(\tilde{r}\right)$ with fixed $n=2.5$ are plotted. In the right graph (b), the $n$ dependence of $V_{\text{eff}}\left(\tilde{r}\right)$ with fixed $\mathcal{B}=0$ is shown. For each value of $n$, there is a critical value of $\mathcal{B}$ which makes the ground-state energy of Hamiltonian (22) exactly zero. In the region of $\mathcal{B}$ shown in the left graph (a), the ground-state eigenvalues are negative.

Figure 6

The critical value ${\mathcal{B}}_{\text{c}}$ as a function of $n$. The values of ${\mathcal{B}}_{\text{c}}$ are plotted on logarithmic scale. We find that ${\mathcal{B}}_{\text{c}}$ increases with $n$. Eventually, in the limit $n\to 3$, ${\mathcal{B}}_{\text{c}}$ diverges.

Figure 7

The $n$ dependence of the number of the trapped bosons in $B={\tilde{U}}^{-1}{\mathcal{B}}_{\text{c}}$ for a given $n$. It has the minimal value of $\sim {10}^4$ around $n=2.25$.

Figure 8

The example of the trap potential $\tilde{V}\left(\tilde{r}\right)$ reproducing the skyrmion. The density profile under this potential is given by Eq. (66) with $n=2.25$ and $\mathcal{B}=213.495$. As seen in Fig. 7, for these $n$ and $\mathcal{B}$, the number of the trapped bosons is minimal $\left(\sim {10}^4\right)$.